Compton scattering

From Wikipedia, the free encyclopedia

In physics, Compton scattering or the Compton effect, is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. Inverse Compton scattering also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength increases by is called the Compton shift. Although nuclear compton scattering exists, what is meant by Compton scattering usually is the interaction involving only the electrons of an atom. Compton effect was observed by Arthur Holly Compton in 1923 and further verified by his graduate student Y. H. Woo in the years followed. Arthur Compton earned the 1927 Nobel Prize in Physics for the discovery.

The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of charged particles scattered by an electromagnetic wave, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency.

The interaction between high energy photons and electrons results in the electron being given part of the energy (making it recoil), and a photon containing the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. If the photon still has enough energy left, the process may be repeated.

Compton scattering occurs in all materials and predominantly with photons of medium energy, i.e. about 0.5 to 3.5 MeV. It is also observed that high-energy photons; photons of visible light or higher frequency, for example, have sufficient energy to even eject the bound electrons from the atom (Photoelectric effect).

1 The Compton shift formula
- 1.1 Derivation
2 Applications
- 2.1 Compton scattering
- 2.2 Inverse Compton scattering
3 See also
4 External links

[edit] The Compton shift formula

For differential cross section of Compton scattering, see Klein-Nishina formula.

Compton Scattering (in the rest frame of the target)

Compton used a combination of three fundamental formulas representing the various aspects of classical and modern physics, combining them to describe the quantum behavior of light.

Light as a particle, as noted previously in the photoelectric effect.
Relativistic dynamics Special Theory of Relativity
Trigonometry - Law of cosines

The final result gives us the Compton scattering equation:

<math> \lambda_2 = \frac{h}{m_e c}(1-\cos{\theta}) + \lambda_1 </math>

where

<math>\lambda_1 </math> is the wavelength of the photon before scattering,

<math>\lambda_2 </math> is the wavelength of the photon after scattering,

m_e is the mass of the electron,

h/(m_ec) is known as the Compton wavelength,

θ is the angle by which the photon's heading changes,

h is Planck's constant, and

c is the speed of light.

Collectively, the Compton wavelength is 2.43×10^-12 meter.

[edit] Derivation

Begin with energy and momentum conservation:

<math>E_\gamma + E_e = E^\prime_{\gamma^\prime} + E^\prime_{e^\prime} \quad \quad (1) \,</math>

<math>\vec p_\gamma = \vec{p}_{\gamma^\prime} + \vec{p}_{e^\prime} \quad \quad \quad \quad \quad (2) \,</math>

where

<math>E_\gamma \,</math> and <math>p_\gamma \,</math> are the energy and momentum of the photon and

<math>E_e \,</math> and <math>p_e \,</math> are the energy and momentum of the electron.

[edit] Solving (1)

Now we fill in for the energy part:

<math>E_{\gamma} + E_{e} = E_{\gamma'} + E_{e'}\,</math>

We solve this for p_e':

[edit] Solving (2)

Rearange equation (2)

<math>\vec{p}_{e'} = \vec{p}_\gamma - \vec{p}_{\gamma'} \,</math>

and square it to see

<math>{\vec{p_{e'}}}^2 = {(\vec{p_{\gamma}} - \vec{p_{\gamma'}})}^2</math>

<math>{\vec{p_{e'}}}^2 = \vec{p_\gamma}^2 + \vec{p_{\gamma'}}^2 - 2\vec{p}_\gamma \cdot \vec{p}_{\gamma'} \, </math>

<math>{\vec{p_{e'}}}^2 = \vec{p_\gamma}^2 + \vec{p_{\gamma'}}^2 - 2|p_{\gamma}||p_{\gamma'}|\cos(\theta) \,</math>

<math>p_{e'}^2 = \left(\frac{h f}{c}\right)^2 + \left(\frac{h f'}{c}\right)^2 - 2\left( \frac{hf}{c} \right) \left(\frac{h f'}{c} \right) \cos{\theta} \quad \quad \quad (4) </math>

[edit] Putting it together

Then we have two equations for <math>p_{e'}^2</math> (eq 3 & 4), which we equate:

<math> \left(\frac{h f}{c}\right)^2 + \left(\frac{h f'}{c}\right)^2 - \frac{2h^2 ff'\cos{\theta}}{c^2} = \frac{(hf + mc^2-hf')^2 -m^2c^4}{c^2} \,</math>

Now, one simplifies. First by multiplying both sides by c²:

<math>h^2 f^2 + h^2 f'^2 - 2h^2 ff' \cos \theta = (hf + mc^2 - hf')^2 - m^2c^4 . \,</math>

Next, multiply out the right-hand side:

<math>h^2f^2+h^2f'^2-2h^2ff'\cos{\theta} = h^2f^2+m^2c^4+h^2f'^2-2h^2ff'+2h(f-f')mc^2 -m^2c^4 .\,</math>

A few terms cancel from both sides, so we have

<math> -2h^2ff'\cos{\theta} = -2h^2ff'+2h(f-f')mc^2 .\,</math>

Then divide both sides by <math>2h</math> to see

<math>hff'\cos{\theta} = hff'-(f-f')mc^2 \,</math>

<math>(f-f')mc^2 = hff'(1-\cos{\theta}) .\,</math>

Now divide both sides by <math>mc^2</math> and then by <math>ff^\prime</math>:

<math>\frac{f-f^\prime}{f f^\prime} = \frac{h}{mc^2}\left(1-\cos \theta \right) . \,</math>

Now the left-hand side can be rewritten as simply

<math> \frac{1}{f^\prime} - \frac{1}{f} = \frac{h}{mc^2}\left(1-\cos \theta \right) \,</math>

This is equivilent to the Compton scattering equation, but it is usually written using <math>\lambda</math>'s rather than <math>f</math>'s. To make that switch use

<math>f=\frac{c}{\lambda} \,</math>

so that finally,

<math>\lambda'-\lambda = \frac{h}{mc}(1-\cos{\theta}) \, </math>

[edit] Applications

[edit] Compton scattering

Compton scattering is of prime importance to radiobiology, as it happens to be the most probable interaction of high energy X rays with atomic nuclei in living beings and is applied in radiation therapy.

Compton scattering has on occasion been proposed as an alternative explanation for the phenomenon of the redshift by opponents of the Big Bang theory, although this is not generally accepted because the influence of the Compton scattering would be noticeable in the spectral lines of distant objects and this is not observed.

In material physics, Compton scattering can be used to probe the wave function of the electrons in matter in the momentum representation.

Compton Scatter is an important effect in Gamma spectroscopy which gives rise to the Compton edge, as it is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect.

[edit] Inverse Compton scattering

Inverse Compton scattering is important in astrophysics. In X-ray astronomy, the accretion disk surrounding a black hole is believed to produce a thermal spectrum. The lower energy photons produced from this spectrum are scattered to higher energies by relativistic electrons in the surrounding corona. This is believed to cause the power law component in the X-ray spectra (0.2-10 keV) of accreting black holes.

The effect is also observed when photons from the Cosmic microwave background move through the hot gas surrounding a galaxy cluster. The CMB photons are scattered to higher energies by the electrons in this gas, resulting in the Sunyaev-Zel'dovich effect.

[edit] See also

‹ The template below has been proposed for deletion. See templates for deletion to help reach a consensus on what to do. ›

Quantum electrodynamics v • d • e</div>
electron \| positron \| photon self-energy \| vacuum polarization \| vertex function Gupta-Bleuler formalism \| ξ gauge \| Ward identities Compton scattering \| Bhabha scattering \| Moeller scattering anomalous magnetic dipole moment bremsstrahlung \| positronium